Approximation by integro cubic splines

Years ago, after I presented a paper on spline functions at Oxford University, Professor Powell criticized us for using, most of the time, the function values rather than the integral values on constructing of the spline functions. His comments and his request became the main motivation for this work. In this paper, we assume that, on each subinterval of the spline interval [a, b], the integral value of the function y = y(x) is known. By using these values, rather than the function values at the knots, we introduce a class of new types of interpolatory cubic splines to approximate the function y = y(x). The selection of the end conditions for our integro cubic splines will be discussed. The numerical examples and computational results, illustrate and guarantee a higher accuracy for this approximation.